Limit of cos(1/x) as x approaches 0

The limit of cos(1/x) as x approaches 0 does not exist. In this post, we will learn how to find the limit of cos(1/x).

lim_x→0 cos(1/x) Does Not Exist.

Limit of cos(1/x) when x→0

Question: What is the limit cos(1/x) when x tend to 0?

Answer: The limit of cos(1/x) does not exist when x tends to 0.

Explanation:

The following theorem will be used to find the limit of cos(1/x) when x tends to 0.

Theorem: If limx→c f(x) = L, then for every sequence xn→c we always have limn→∞ f(xn) = L.

Let

x_n = $\dfrac{1}{2n \pi}$

y_n = $\dfrac{1}{(2n+1) \pi}$.

Note that when n tends to ∞, the above two sequences

x_n, y_n →0.

But,

lim_n→∞ f(x_n) = lim_n→∞  cos 2nπ = 1.

lim_n→∞ f(y_n) = lim_n→∞  cos (2n+1)π = -1.

So by the above theorem, we conclude that the limit of cos(1/x) when x tends to 0 does not exist.

Also Read: Limit of sin(1/x) as x approaches 0

Limit of sinx/x when x→∞ | Limit of (cosx-1)/x when x→0

Limit of sin(x²)/x when x→0 | Limit of sin(√x)/x when x→0

Limit of log(1+x)/x when x→0 | Limit of (e^x-1)/x when x→0

Limit of (1+$\frac{1}{n})ⁿ when n→∞

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